Mathematics 239 :  Introduction to Mathematical Proof
Fall 2023
Introduction

Professor:        Jeff Johannes                                    Section 2    MWF    1:30 - 2:20p    Fraser 119
Office:             South 326A
Telephone:       245-5403
Office Hours:   Monday 5-6p in South 309, Tuesday 1-2p in South 309. Wednesday 12:30-1:20p in South 309, Thursday 8-9p in South 336, Friday 2:30-3:30p in South 309, and by appointment or visit.
Email Address: Johannes@Geneseo.edu
Web-page:        http://www.geneseo.edu/~johannes

### Textbooks

Mathematics:  A Discrete Introduction, Third Edition, Edward R. Scheinerman (link to chapter 1)

### Purposes

• to develop familiarity and comfort with the language of more formal mathematics and proofs before taking upper division mathematics coursework
• to learn and justify several techniques for counting

### Overview

It is often said that mathematics is a language.  In this class you will begin to learn to speak this language.  Just like in an introductory language course, we will start with the most fundamental concepts and grammar rules.  After we have some familiarity with the language of formal mathematics, we will practice this language in the setting of counting problems of different types.  More like an advanced language class, merely memorizing the vocabulary will not suffice (in fact, hopefully we can keep vocabulary to a minimum), but rather you will be required to understand and speak clearly in this language.  The material learned here will help you understand the mathematics you read and clarify the mathematics you write.  Because we are learning how to write mathematics, exposition will also be a component in your evaluation.

### Reading

I have intentionally chosen a very readable text.  In addition to planning time to do homework, please take time to carefully read the sections in the book.  Notice use of the words “time" and “carefully".  Read the sections slowly.  As the author indicates in the preface, read actively.  If you do not understand some statement reread it, think of some potential meanings and see if they are consistent, and if all else fails, ask me.  If you do not believe a statement, check it with your own examples.  Finally, if you understand and believe the statements, consider how you would convince someone else that they are true, in other words, how would you prove them?
Because the text is exceptionally accessible, we will structure class-time more as an interactive discussion of the reading than lecture.  For each class day there is an assigned reading.  Read and take notes on the section before coming to class.  In addition to the reading, there are also indicated exercises to check that you understood the reading.  When we complete questions from the reading we will discuss those indicated exercises during the class discussion.

### Learning Outcomes

Upon successful completion of Math 239 a student will be able to
• Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments,
• Perform set operations on finite and infinite collections of sets and be familiar with properties of set operations,
• Determine equivalence relations on sets and equivalence classes,
• Work with functions and in particular bijections, direct and inverse images and inverse functions,
• Construct direct and indirect proofs and proofs by induction and determine the appropriateness of each type in a particular setting. Analyze and critique proofs with respect to logic and correctness, and
• Unravel abstract definitions, create intuition-forming examples or counterexamples, and prove conjectures.
• Write solutions to problems and proofs of theorems that meet rigorous standards based on content, organization and coherence, argument and support, and style and mechanics.

### Grading

Your grade in this course will be based upon your performance on homework, quizzes, one colloquium report, and three exams.  The weight assigned to each is designated below:
Homework (6)                5% each
Quizzes (3)                     5% each
Colloquium Report (1)   5%
In-class exams (2)        15% each
Final exam (1)             20%

### Problem Sets

There are problem sets for each chapter.  They will be due shortly after each chapter ends.  You are encouraged to consult with me outside of class on any questions toward completing the homework.  You are also encouraged to work together on homework assignments, but each must write up their own well-written solutions.  A good rule for this is it is encouraged to speak to each other about the problem, but you should not read each other's solutions.  A violation of this policy will result in a zero for the entire assignment and reporting to the Dean of Students for a violation of academic integrity.  Each question will be counted in the following manner:
0 - missing or plagiarised question
1 - question copied
2 - partial question
3 - completed question (with some solution)
4 - completed question correctly and well-written
Each entire homework set will then be graded on a 90-80-70-60% (decile) scale.  Late items will not be accepted, as solutions will be posted immediately.  Homework will be returned on the following class day.  Please feel free to discuss any homework with me outside of class or during review.

### Solutions and Plagiarism

There are plenty of places that one can find all kinds of solutions to problems in this class.  Reading them and not referencing them in your work is plagiarism, and will be reported as an academic integrity violation.  Reading them and referencing them is not quite plagiarism, but does undermine the intent of the problems.  Therefore, if you reference solutions you will receive 0 points, but you will *not* be reported for an academic integrity.  Simply - please do not read any solutions for problems in this class.  Any work written, developed, or created, in whole or in part, by generative artificial intelligence (AI) is considered plagiarism and will not be tolerated. While the ever-changing developments with AI will find their place in our workforces and personal lives, in the realm of education and learning, this kind of technology does not help us achieve our educational goals. The use of AI prevents the opportunity to learn from our experiences and from each other, to play with our creative freedoms, to problem-solve, and to contribute our ideas in authentic ways. Geneseo is a place for learning, and this class is specifically a space for learning how to advance our thinking and professional practice. AI cannot do that learning for us.

### Opening Meeting

Students will earn two extra points on the first problem set by visiting office hours during the first two weeks of classes, i.e. no later than 11 September.

### Presentations

When discussing the new material for each section, each day will be begin with an opportunity for student presentation.  There are presentation problems listed on the reading schedule.  Students will earn one extra point for the corresponding problem set by attempting a presentation and two extra points by presenting well.  I will have a priority list at all times for presentations.  Students may present more than once per chapter, but representations have lowest priority.  Students may earn no more than two extra points in this fashion, and may only earn two points by doing a presentation well, not by presenting poorly twice.

### Quizzes

There will be short quizzes after the homework has been returned, covering the material in the chapter from the homework.  For chapters immediately preceding exams, there will be no quiz.  Quizzes will consist of routine questions, and will have limited opportunity for partial credit. Because quizzes will consist of routine questions, they will be graded on a decile scale. There will be no makeup quizzes.

### Colloquium Report

Attend one of the department colloquium talks.  Write a report.  In the report, describe the content of the talk (including a detailed discussion of the mathematics).  In addition to your description of the talk, also write how this talk added to your understanding of the nature of mathematics.  Papers are due within a classweek of the colloquium presentation.  I will gladly look at papers before they are due to provide comments.

### Exams

There will be two exams during the semester and a final exam during finals week.  If you must miss an exam, it is necessary that you contact me before the exam begins.  Exams require that you show ability to solve unfamiliar problems and to understand and explain mathematical concepts clearly.  The bulk of the exam questions will involve problem solving and written explanations of mathematical ideas.  The final exam will be half an exam focused on the final two chapters, and half a cumulative exam.  Exams will be graded on a scale approximately (to be precisely determined by the content of each individual exam) given by
100 - 80%   A
79 - 60%    B
59 - 40%    C
39 - 20%    D
below 20%   E
For your interpretive convenience, I will also give you an exam grade converted into the decile scale.  The exams will be challenging and will require thought and creativity.  They will not include filler questions (hence the full usage of the grading scale).

### Feedback

Occasionally you will be given anonymous feedback forms.  Please use them to share any thoughts or concerns for how the course is running.  Remember, the sooner you tell me your concerns, the more I can do about them.  I have also created a web-site which accepts anonymous comments.  If we have not yet discussed this in class, please encourage me to create a class code.  This site may also be accessed via our course page on a link entitled anonymous feedback.  Of course, you are always welcome to approach me outside of class to discuss these issues as well.

### Social Psychology

Wrong answers are important.  We as individuals learn from mistakes, and as a class we learn from mistakes.  You may not enjoy being wrong, but it is valuable to the class as a whole - and to you personally.  We frequently will build correct answers through a sequence of mistakes.  I am more impressed with wrong answers in class than with correct answers on paper.  I may not say this often, but it is essential and true.  Think at all times - do things for reasons.  Your reasons are usually more interesting than your choices.  Be prepared to share your thoughts and ideas.  Perhaps most importantly "No, that's wrong." does not mean that your comment is not valuable or that you need to censor yourself.  Learn from the experience, and always try again.  Don't give up.

### Accessibility Accommodations

SUNY Geneseo is dedicated to providing an equitable and inclusive educational experience for all students. The Office of Accessibility (OAS) will coordinate reasonable accommodations for persons with disabilities to ensure equal access to academic programs, activities, and services at Geneseo.  Students with approved accommodations may submit a semester request to renew their academic accommodations. Please visit the OAS website for information on the process for requesting academic accommodationsContact the OAS by email, phone, or in-person:  Office of Accessibility Services Erwin Hall 22 585-245-5112 access@geneseo.edu

### Religious Holidays

It is my policy to give students who miss class because of observance of religious holidays the opportunity to make up missed work.  You are responsible for notifying me no later than September 10 of plans to observe the holiday.

### Schedule (subject to change)

August 28
- September 11    Chapter I reading discussions
September 13    Chapter I homework due
September 18   Chapter I quiz taken

September 13-22   Chapter II reading discussions
September 25    Chapter II homework due

September 25-27  review Chapters I and II
September 29         In class exam covering Chapters I and II

October 2-16   Chapter III reading discussions
October 18    Chapter III homework due
October 23    Chapter III quiz taken

October 18-25    Chapter IV reading discussions
October 27    Chapter IV homework due

October 27 - 30    review Chapters III and IV
November 1    In class exam covering Chapters III and IV

November 3 -15    Chapter V reading discussions
November 17    Chapter V homework due
November 27    Chapter V quiz taken

November 17
- December 6    Chapter VII reading discussions
December 8        Chapter VII homework due

December 8-11    review Chapters V and VII, and course as whole

Friday, December 15 3:30-6:30p  Final exam, first half covering chapters V and VII
second half covering course

Problem Sets:

Assignment for Chapter 1:  To the student.1, 3.2, 3.6, 4.3, 4.10, 5.2, 5.12, 5.15, 5.23, 6.2, 6.4, 6.11, 7.8, 7.11

For example, this means that you must complete the second exercise in section 3.  There is also a required exercise at the end of the "to the student" section.

Assignment for Chapter 2:  8.5, 8.12, 8.18, 9.2, 9.8, 10.3, 10.11, 11.5, 12.18, 12.21

Assignment for Chapter 3:  14.6, 14.13, 14.16, 15.6, 15.12, 15.15, 16.11, 16.13, 17.14, 17.23, 17.24, 17.33

Assignment for Chapter 4:  20.5, 20.7, 20.14, 22.9, 22.16 (valued as 6 questions)

Assignment for Chapter 5:  24.4, 24.8, 24.14, 24.16, 24.17, 24.20, 24.22, 25.9, 25.16, 26.4, 26.12, 27.2, 27.13

Assignment for Chapter 7:  35.4, 36.11, 36.15, 37.3, 37.14 (valued as 5 questions), supplemental E-primes.